Mandelbrot set for fractal $n$-gons and zeros of power series

TL;DR

This paper introduces a framework for analyzing the connectedness of zeros of power series with coefficients in finite sets and applies it to prove the connectedness and local connectedness of Mandelbrot sets for fractal n-gons.

Contribution

It establishes a general connection between coefficient graph connectivity and zero set connectedness, and applies this to Mandelbrot sets for fractal n-gons.

Findings

Zeros of power series with connected coefficient graphs are connected in the unit disk.

Mandelbrot sets for fractal n-gons are proven to be connected.

Mandelbrot sets for fractal n-gons are also locally connected.

Abstract

We give a framework to study the connectedness of the set of zeros of power series with coefficients in a finite subset $G\subset \mathbb{C}$. We prove that the set of zeros in the unit disk is connected and locally connected if some graph on the set $G$ of coefficients is connected. Furthermore, we apply this result to the study of the Mandelbrot set $\mathcal{M}_n$ for fractal $n$-gons. We prove that $\mathcal{M}_n$ is connected and locally connected for any $n$.